HAL Id: jpa-00217051
https://hal.archives-ouvertes.fr/jpa-00217051
Submitted on 1 Jan 1977
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
ELECTRONIC STRUCTURE OF SMALL
CUBO-OCTAHEDRAL CLUSTERS OF TRANSITION
METALS
F. Cyrot-Lackmann, M. Desjonquères, M. Gordon
To cite this version:
JOURNAL DE PHYSIQUE Colloque C2, supplément au n° 7, Tome 38, Juillet 1977, page C2-57
ELECTRONIC STRUCTURE OF SMALL CUBO-OCTAHEDRAL CLUSTERS
OF TRANSITION METALS
F . C Y R O T - L A C K M A N N , M . C . D E S J O N Q U E R E S a n d M . B . G O R D O N G r o u p e d e s T r a n s i t i o n s d e P h a s e s , C . N . R . S . B . P . 166, 38042 G r e n o b l e C e d e x , F r a n c e
Résumé. — La structure électronique de petits amas cubo-octaédriques de métaux de transition c.f.c. de taille croissante jusqu'à 1 289 atomes est décrite au moyen de densités d'états locales (LDS) en différents sites du cristal dans le cadre de l'approximation des liaisons fortes associée à la méthode des moments. Certaines caractéristiques des LDS sont gouvernées par l'environnement de premiers voisins, comme on le voit quand on compare avec des calculs effectués sur l'arête saillante de surfaces avec marches ayant le même environnement local, et avec des résultats en volume. Mais leurs structures fines dépendent fortement des couches de voisins plus distantes. L'énergie de cohésion est calculée en fonction du remplissage de la bande pour des amas de tailles différentes, et on montre qu'elle converge vite vers celle du volume.
Abstract. — The electronic structure of F.C.C. transition metal small cubo-octahedral clusters of increasing size up to 1 289 atoms is described by means of local densities of states (LDS) on various crystal sites in the framework of the tight binding approximation associated with the moment method. The general trends of the LDS are governed by the nearest neighbour shells, as appears when comparing with calculations at the protruding edge of stepped surfaces having the same local environment, and with bulk results. But some outstanding details are strongly dependent on more distant shells. Cohesive energy is calculated in terms of band filling for different cluster sizes and shows to converge quickly to that of the bulk.
1. Introduction. — I t is well b e l i e v e d t h a t catalytic p r o p e r t i e s of transition m e t a l s d e p e n d o n t h e size of their crystallites. A n u n d e r s t a n d i n g of their electronic s t r u c t u r e c a n surely h e l p t o s h e d s o m e light o n this p h e n o m e n o n . T w o b a s i c q u e s -t i o n s arise a -t -this level w h e n changing -t h e size a n d s h a p e of t h e t r a n s i t i o n m e t a l clusters : h o w d o e s their e l e c t r o n i c s t r u c t u r e c h a n g e , a n d a r e w e able t o p r e d i c t their stability ?
Only a f e w studies h a v e y e t b e e n d o n e o n t h e e l e c t r o n i c s t r u c t u r e of small transition metal p a r t i
-c l e s . s-p-d e n e r g y levels of -clusters u p t o 55 a t o m s h a v e b e e n studied w i t h m e t h o d s issued f r o m t h e o r e t i c a l c h e m i s t r y [ 1 , 2 , 3] or self c o n s i s t e n t field X a s c a t t e r e d w a v e m e t h o d [4, 5]. N o c o m p a r i s o n of e l e c t r o n i c densities of s t a t e s b e t w e e n c l u s t e r s of different sizes or g e o m e t r i e s h a v e y e t b e e n r e p o r -t e d . A n y firm c o n c l u s i o n a b o u -t -t h e general -t r e n d s of t h e s e densities of s t a t e s of small c l u s t e r s , particularly in v i e w t o t h o s e of bulk or semi-infinite c r y s t a l s , c a n n o t t h e n b e d r a w n , a n d s o m e c o n t r o -v e r s y still e x i s t s . M o r e o -v e r , t h e relati-ve stability of t h e s e c l u s t e r s h a s only b e e n studied with semi-empirical pair potential m o d e l s [6, 7, 8 ] .
I n t h e p r e s e n t p a p e r , t h e e l e c t r o n i c s t r u c t u r e of increasing size, u p t o 1 289 a t o m s , c u b o - o c t a h e d r a l c l u s t e r s of f . c . c . transition m e t a l s , is d e s c r i b e d b y m e a n s of t h e local d e n s i t y of s t a t e s ( L D S ) o n different s i t e s , a n d their c o h e s i v e e n e r g y calculat e d , in calculat h e f r a m e w o r k of calculat h e calculatighcalculat binding a p p r o x i -m a t i o n a s s o c i a t e d w i t h t h e -m o -m e n t -m e t h o d . A c o m p a r i s o n is m a d e with t h e L D S o n sites of s t e p p e d s u r f a c e s w i t h t h e s a m e first n e a r e s t n e i g h b o u r e n v i r o n m e n t . Only d-states h a v e b e e n t a k e n i n t o a c c o u n t a s a first a p p r o a c h , a n d s-d hybridization neglected — a s in b u l k a n d s u r f a c e calculations — since transition m e t a l s h a v e a strong d c h a r a c t e r in their v a l e n c e s t a t e s [9]. T h e effect of including s b a n d w o u l d b e essentially t o modify t h e L D S n e a r t h e b a n d e d g e s [10]. Calculations w e r e m a d e n o t self c o n s i s t e n t l y . W e e x p e c t t h a t self c o n s i s t e n c e w o u l d shift t h e L D S p e a k s t o higher or t o l o w e r energies d e p e n d i n g o n w h e t h e r t h e b a n d is m o r e o r less t h a n half-filled, b u t w i t h o u t changing its general s h a p e [11].
R e s u l t s s h o w t h a t t h e fine s t r u c t u r e s of t h e L D S a r e strongly d e p e n d e n t o n cluster size, b u t t h a t a general t r e n d exists t o w a r d s t h e L D S of b u l k or surface sites having t h e s a m e n e a r e s t n e i g h b o u r e n v i r o n m e n t .
2. The method. — W e j u s t recall its p r i n c i p l e s , a l r e a d y d e s c r i b e d e l s e w h e r e [9, 12, 13]. W e s t a r t from a o n e - e l e c t r o n tight-binding hamiltonian :
5 f = T + E V i CD
i
w h e r e T is t h e kinetic e n e r g y a n d V-, t h e potential energy c e n t e r e d on site i.
C2-58 F. CYROT-LACKMANN, M. C. DESJONQUERES AND M. B. GORDON
We will study the local density of states (LDS) on
a site i defined as :
where
1
iA)
is the atomic orbital and
In)the
eigenfunction of the hamiltonian
X
with the corres-
ponding eigenvalue En, and niA(E) is the contribu-
tion of the
I
iA )orbital to the LDS.
The total density of states n(E) then writes :
where N is the number of atoms.
The moments of the LDS are defined by
:The LDS ni(E) is built up from its first moments
using the continuous fraction expansion of its
Hilbert transform, namely, the Green function [13]
which can be written as :
where the a,, b, coefficients are given through the
2
pfirst moments of the LDS. These coefficients
are usually converging very quickly towards their
asymptotic limit giving the energy spectrum limits
as shown in many applications for bulk or semi-
infinite crystals [ l l , 14, 151. But, in the cluster
problem, we are faced with some peculiar difficul-
ties. Indeed, the density of states being strictly a
finite series of delta functions, the continuous
fraction expansion becomes a finite series too, with
a number of terms equal to the number of different
energy levels of the system [14]. The coefficients
thus are converging to a zero limit (Fig. 1).
However, the moment method do give good results,
as will be shown later on by comparison with an
exact diagonalization of the hamiltonian. The
moment method will then be of peculiar use for
large clusters for which exact solutions are not
available.
3. Cubo-octahedral clusters
-
Geometric featu-res.
-
We have studied the electronic structure of
cubo-octahedral f.c.c. clusters described by Van
Hardeveld and Hartog'[16], which are truncated
octahedra, and therefore exhibit six square (100)
faces and eight hexagonal (111) ones, all having
equal edge lengths (Fig. 2).
Corner
tzJ
FIG. 1. - The bi coefficients for m = 2 and m = 3 cubo- octahedral clusters.
FIG. 2. - The cubo-octahedron.
The cluster size can be defined by m, the number
of atoms on an edge, the smallest being m
=2.
They can be grouped into two sequences depending
on whether they do have a central atom (m odd) or
not (m even). For a given cluster size, all the corners
are equivalent, as they lay at the intersection of two
(I 11)
planes with a (100) one. There are two sorts of
edges, namely, between two hexagonal faces or
between an hexagonal and
a square face.
ELECTRONIC STRUCTURE OF TRANSITION METAL CLUSTERS C2-59 TABLE
I
Size and number o f different atoms for f.c.c.
cubo-octahedral clusters
Diameter m (A)NT
NB Ns Nc N E MINT - --
-
-
-
-
- 2 7.5 38 6 32 24-
0.84 3 15.0 201 79 122 24 12 0.61 4 22.5 586 314 272 24 24 0.46 5 30.0 1289 807 482 24 36 0.37 m = number of atoms on an edge.Diameter : calculated for an interactomic distance of 2.5
A
(Ni). NT = total number of atoms.NB =number of bulk atoms (having a complete nearest neighbour shell).
Ns = number of surface atoms. N, = number of corner atoms.
NE = number of atoms at the edge between an hexagonal and a square face.
4. Results
and
discussion.
-
All our results were
obtained with the tight binding parameters of
paramagnetic nickel, taking into account only
nearest neighbour matrix elements, as in f .c.c.
structures those between not-nearest neighbours
are negligible [l 11.
We solved the exact eigenvalue problem for the
m
=2 cubo-octahedron. Diagonalization of the
190
x
190 hamiltonian giving 79 poles was achieved
to test the results of the moments method (Fig. 3).
A continuous total density of states was calculated
using a gaussian broadening of the discrete
spectrum :
each level entering with a weight
w,equal to its
degeneracy, and
u =0.003 Ryd. Continuous LDS at
broaden, ng of e r a s t l e v e l s
- moment method
- 0 15 - 0 10 - 0 05 0 0 0 05 0 10 0 1 5
E l R y d )
FIG. 3.
-
Total density of states for m = 2 cubo-octahedron. Exact levels with a height proportional t o its degeneracy (vertical lines). A single line has been drawn for very nearby levels which cannot be split at the figure scale. Broadening of exact levels(....), moment method result (-).
desired sites were also generated in this way, but
taking as weight the sum of the square moduli of
the corresponding wave functions at the site of
interest.
The 38 first moments of the LDS on the three
different sites of the
m
=2 cluster, and on the 12
ones for
m
=3 have been calculated and added up
in order to obtain the total densities of states
(Fig. 3, 4). All the densities of states obtained by
the moment method are in excellent agreement
with the exact ones. We see that on increasing the
cluster size, the band width increases, and the
shape of the density of states tends toward the bulk
one, calculated previously [Ill.
FIG. 4. - Total densities of states for m = 2 and m = 3
cubo-octahedra, and bulk f.c.c. crystal.
L.D.S. on the corners for
m
=2, 3, 4 and 5 are
shown on figures
5a and
5b. They present an
alternance on some features like the central peak
which only appears for
m odd. This effect can also
be observed in the total density of states of
figure
2,although not so pronounced, and it is
damped when increasing cluster size.
F. CYROT-LACKMANN, M. C. DESJONQUERES AND M. B. GORDON
m = 2
b r o a d e n q of
exact l e v e l s
- rnornen t rnethaf
FIG. 5a. - LDS on the m = 2 cubo-octahedron corner. Exact
broadened levels (.
.
. .), moment method result (-).Some of the general features of these LDS are
similar, but let us remind that when
m
increases,
there is still only the nearest neighbour environment
which is the same.
We have done a similar study on the edge
between hexagonal and square faces (Fig. 2) of
m
=3,
4and 5 cubo-octahedra. We have found
analogous features for their LDS to those of the
corner (Fig. 7). For
m
=2,
the two atoms on the
edges are in fact at corner sites. We have also done
a comparison with the LDS on the protruding edge
of the [6(111)
x
(OOl)] stepped surface, which has
the same nearest neighbour shell
[15]and similar
trends are found (Fig.
8).RG. 7. - LDS on the edge between hexagonal and square faces
for m = 3, 4 and 5 cubo-octahedra.
1
@ bulkFIG. 5b.
-
LDS on the m = 3, 4 and 5 cubo-octahedra corners.FIG. 6.
-
LDS on the protruding edge of a [9(111) x (0li)jstepped surface (-), bulk f.c.c. crystal (.... ).
FIG. 8. - LDS on the protruding edge of a [6(11 I ) x (01 I)]
stepped surface (-), bulk f.c.c. crystal (.... ).
5. Cohesive
energy.
-
The cohesive energy
per
atom defined by
:E,
= -10
I-:
En (E) d E
was calculated for different Fermi levels. Results
are shown on figure 9 for the cohesive energy
vs.ELECTRONIC STRUCTURE OF TRANSITION METAL CLUSTERS C2-61
FIG. 9. - Cohesive energy vs. band filling for m = 2 (-), m = 2 (-.-.-), bulk f.c.c. crystal (----).
value when every atom is surrounded by all its nearest neighbours.
Let us remark that here also the cohesive energy is well given by a small number of moments. Thus, the ratio of the cohesive energy for a half-filled band for m =
2
and 3, Ec(m = 2)Ec(m = 3) is 0.89, and a
reasoning using a square root of the mean coordina- tion number gives
0.90.
This may be of interest when looking at difference in stability between various cluster shapes (to be published).6. Conclusion.
-
In conclusion, we have seenthat the general features of the electronic structure of small clusters follow, when increasing size, a general trend towards that of bulk or surface sites having the same nearest neighbour environment. Nevertheless, important details of the LDS strongly depend on the size and shape of the clusters far away of the studied site.
It
would then be very interesting to study some other structures of clusters to try to understand the relative influence of the first shell of neighbours and of the symmetry properties of the whole cluster. Studies of binding energy of various adsorbed species could also be engaged fruitfully in connection with the peculiar catalytic properties of these clusters, as the detailed structures of the density of states may play a non-negligible role.References
[I] BAETZOLD, R. C., J. Chem. Phys. 55 (1971) 4363. [9] FRIEDEL, J., The Physics of metals, Ed. J . M. Ziman, [2] BAETZOLD, R. C. and MACK, R. E., J. Chem. Phys. 62 Cambridge (Cambridge University Press) 1%9, p. 340. (1975) 1513. [I01 GASPARD, J. P., HODGES, C. H. and GORDON, M. B., J. [3] BLYHOLDER, G., Surf. Sci. 42 (1974) 249. Phvsiaue Colloa. - - 38 (1977) C2-63. .
,
[4] ROSCH, N. and MENZEL, D., Chem. Phys. 13 (1976) 243. [ll] DESJONQUERES, M. C. and CYROT-LACKMANN, F., J. Phys. [S] MESSMER, R. P., KNUDSON, S. K., JOHNSON, K. H., F 5 (1975) 1368.
DIAMOND, J. B. and YANG, C. J., Phys. Rev. B 13 (1976) CYROT-LACKMANN, F.3 3. Physique Colloq- 31 (1970) C 1-67. 1396. [13] GASPARD, J. P. and CYROT-LACKMANN, F., J. Phys. C 6 [6] CLARK, B. C., HERMAN, R., GAZIS, D. C. and WALLIS, R. (1973) 3077.
F., Ferroelectricity, Edward F. Weller, Editor (Elsevier C141 GASPARD, J. P., ThAse UniversitC de Paris XI, Orsay (1975).
Publishing Company, Amsterdam) 1967, p. 101. [IS] DESJONQUERES, M. C. and CYROT-LACKMANN, F., Solid State Commun. 18 (1976) 1127.
